Answer
$\frac{{25}}{2}$
Work Step by Step
$$\eqalign{
& \int_0^1 {\int_0^{2y} {\left( {9 + 3{x^2} + 3{y^2}} \right)} } dxdy \cr
& = \int_0^1 {\left[ {\int_0^{2y} {\left( {9 + 3{x^2} + 3{y^2}} \right)} dx} \right]} dy \cr
& {\text{Integrate with respect to }}x \cr
& \int_0^{2y} {\left( {9 + 3{x^2} + 3{y^2}} \right)} dx = \left[ {9x + {x^3} + 3x{y^2}} \right]_0^{2y} \cr
& = 9\left( {2y} \right) + {\left( {2y} \right)^3} + 3\left( {2y} \right){y^2} \cr
& = 18y + 8{y^3} + 6{y^3} \cr
& = 18y + 14{y^3} \cr
& \int_0^1 {\left[ {\int_0^{2y} {\left( {9 + 3{x^2} + 3{y^2}} \right)} dx} \right]} dy = \int_0^1 {\left( {18y + 14{y^3}} \right)} dy \cr
& {\text{Integrating}} \cr
& = \left[ {9{y^2} + \frac{7}{2}{y^4}} \right]_0^1 \cr
& = 9{\left( 1 \right)^2} + \frac{7}{2}{\left( 1 \right)^4} \cr
& = \frac{{25}}{2} \cr} $$
